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Theorem dvlog2lem 20412
Description: Lemma for dvlog2 20413. (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
dvlog2.s  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
Assertion
Ref Expression
dvlog2lem  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )

Proof of Theorem dvlog2lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvlog2.s . . . . 5  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
2 cnxmet 18680 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3 ax-1cn 8983 . . . . . 6  |-  1  e.  CC
4 1re 9025 . . . . . . 7  |-  1  e.  RR
54rexri 9072 . . . . . 6  |-  1  e.  RR*
6 blssm 18344 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1279 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3323 . . . 4  |-  S  C_  CC
98sseli 3289 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
103subid1i 9306 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 10648 . . . . . . . . . . . 12  |-  -oo  e.  RR*
12 0re 9026 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 10924 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (  -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 654 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  RR
1514sseli 3289 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  RR )
1612a1i 11 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  0  e.  RR )
174a1i 11 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR )
18 elioc2 10907 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) ) )
1911, 12, 18mp2an 654 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) )
2019simp3bi 974 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 9577 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4189 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 8982 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3302 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  CC
2524sseli 3289 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2389 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 18678 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
283, 25, 27sylancr 645 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
29 0le1 9485 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9161 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 12163 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2421 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4181 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 18679 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3635a1i 11 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( Met `  CC ) )
373a1i 11 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  CC )
38 metcl 18273 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1184 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9089 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
414, 39, 40sylancr 645 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4234, 41mpbid 202 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  ( 1 ( abs 
o.  -  ) x
)  <  1 )
432a1i 11 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
445a1i 11 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR* )
45 elbl2 18326 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  RR* )  /\  ( 1  e.  CC  /\  x  e.  CC ) )  -> 
( x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4643, 44, 37, 25, 45syl22anc 1185 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4742, 46mtbird 293 . . . . 5  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4847con2i 114 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  (  -oo (,] 0
) )
4948, 1eleq2s 2481 . . 3  |-  ( x  e.  S  ->  -.  x  e.  (  -oo (,] 0 ) )
509, 49eldifd 3276 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  (  -oo (,] 0 ) ) )
5150ssriv 3297 1  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    \ cdif 3262    C_ wss 3265   class class class wbr 4155    o. ccom 4824   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    -oocmnf 9053   RR*cxr 9054    < clt 9055    <_ cle 9056    - cmin 9225   (,]cioc 10851   abscabs 11968   * Metcxmt 16614   Metcme 16615   ballcbl 16616
This theorem is referenced by:  dvlog2  20413  logtayl  20420  logtayl2  20422  efrlim  20677  lgamcvg2  24620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-xadd 10645  df-ioc 10855  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-xmet 16621  df-met 16622  df-bl 16623
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